# Unsupervised Learning of Full-Waveform Inversion: Connecting CNN and Partial Differential Equation in a Loop

arXiv: https://arxiv.org/abs/2110.07584 · HTML: https://mlsanity.com/paper/2110.07584/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 36 references that are in the map.
- First posted on arXiv 14 Oct 2021.

## Builds on (most cited first, 12 of 12)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFs](https://mlsanity.com/paper/1811.07875/) (2018)
- [Deep-learning inversion: a next generation seismic velocity-model building method](https://mlsanity.com/paper/1902.06267/) (2019)
- [Stochastic seismic waveform inversion using generative adversarial networks as a geological prior](https://mlsanity.com/paper/1806.03720/) (2018)
- [Deep-Learning Inversion of Seismic Data](https://mlsanity.com/paper/1901.07733/) (2019)
- [Data-driven Seismic Waveform Inversion: A Study on the Robustness and Generalization](https://mlsanity.com/paper/1809.10262/) (2018)
- [Physics-Consistent Data-driven Waveform Inversion with Adaptive Data Augmentation](https://mlsanity.com/paper/2009.01807/) (2020)
- [Full waveform inversion guided by travel time tomography](https://mlsanity.com/paper/1607.00968/) (2016)

## Built on by (most cited first, 29 of 29)

- [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023)
- [OpenFWI: Large-Scale Multi-Structural Benchmark Datasets for Seismic Full Waveform Inversion](https://mlsanity.com/paper/2111.02926/) (2021)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Solving Seismic Wave Equations on Variable Velocity Models with Fourier Neural Operator](https://mlsanity.com/paper/2209.12340/) (2022)
- [Knowledge-guided Machine Learning: Current Trends and Future Prospects](https://mlsanity.com/paper/2403.15989/) (2024)
- [WISE: full-Waveform variational Inference via Subsurface Extensions](https://mlsanity.com/paper/2401.06230/) (2023)
- [An Empirical Study of Large-Scale Data-Driven Full Waveform Inversion](https://mlsanity.com/paper/2307.15388/) (2023)
- [An Intriguing Property of Geophysics Inversion](https://mlsanity.com/paper/2204.13731/) (2022)
- [Auto-Linear Phenomenon in Subsurface Imaging](https://mlsanity.com/paper/2305.13314/) (2023)
- [A Unified Framework for Forward and Inverse Problems in Subsurface Imaging using Latent Space Translations](https://mlsanity.com/paper/2410.11247/) (2024)
- [Foundation Models for Environmental Science: A Survey of Emerging Frontiers](https://mlsanity.com/paper/2504.04280/) (2025)
- [Inversion of Time-Lapse Surface Gravity Data for Detection of 3D CO2 Plumes via Deep Learning](https://mlsanity.com/paper/2209.02850/) (2022)
- [Extremely Weak Supervision Inversion of Multi-physical Properties](https://mlsanity.com/paper/2202.01770/) (2022)
- [Edge-InversionNet: Enabling Efficient Inference of InversionNet on Edge Devices](https://mlsanity.com/paper/2310.09667/) (2023)
- [Integrating Physics of the Problem into Data-Driven Methods to Enhance Elastic Full-Waveform Inversion with Uncertainty Quantification](https://mlsanity.com/paper/2406.05153/) (2024)
- [SG-DeepONet: Source-generalized deep operator learning for full waveform inversion](https://mlsanity.com/paper/2408.08005/) (2024)
- [Leveraging Deep Operator Networks (DeepONet) for Acoustic Full Waveform Inversion (FWI)](https://mlsanity.com/paper/2504.10720/) (2025)
- [Unveiling the Mechanism of Continuous Representation Full-Waveform Inversion: A Wave Based Neural Tangent Kernel Framework](https://mlsanity.com/paper/2603.22362/) (2026)
- [On the Robustness and Generalization of Deep Learning Driven Full Waveform Inversion](https://mlsanity.com/paper/2111.14220/) (2021)
- [Overview of Physics-Informed Machine Learning Inversion of Geophysical Data](https://mlsanity.com/paper/2310.08109/) (2023)
- [SVInvNet: A Densely Connected Encoder-Decoder Architecture for Seismic Velocity Inversion](https://mlsanity.com/paper/2312.08194/) (2023)
- [Physics-guided Full Waveform Inversion using Encoder-Solver Convolutional Neural Networks](https://mlsanity.com/paper/2405.17696/) (2024)
- [Survey of Deep Learning and Physics-Based Approaches in Computational Wave Imaging](https://mlsanity.com/paper/2410.08329/) (2024)
- [On a Hidden Property in Computational Imaging](https://mlsanity.com/paper/2410.08498/) (2024)
- [SA-EMO: Structure-Aligned Encoder Mixture of Operators for Generalizable Full-waveform Inversion](https://mlsanity.com/paper/2511.11627/) (2025)
- [Improving Full Waveform Inversion in Large Model Era](https://mlsanity.com/paper/2603.00377/) (2026)
- [SPAMoE: Spectrum-Aware Hybrid Operator Framework for Full-Waveform Inversion](https://mlsanity.com/paper/2604.07421/) (2026)
- [Deciphering Neural Reparameterized Full-Waveform Inversion with Neural Sensitivity Kernel and Wave Tangent Kernel](https://mlsanity.com/paper/2605.14370/) (2026)
- [IQ-JEPA: A Joint-Embedding Predictive Architecture with a Hermitian Vision Transformer for Sound Speed and Attenuation Estimation from Ultrasound IQ Data](https://mlsanity.com/paper/2607.22351/) (2026)

## Abstract (arXiv)

This paper investigates unsupervised learning of Full-Waveform Inversion (FWI), which has been widely used in geophysics to estimate subsurface velocity maps from seismic data. This problem is mathematically formulated by a second order partial differential equation (PDE), but is hard to solve. Moreover, acquiring velocity map is extremely expensive, making it impractical to scale up a supervised approach to train the mapping from seismic data to velocity maps with convolutional neural networks (CNN). We address these difficulties by integrating PDE and CNN in a loop, thus shifting the paradigm to unsupervised learning that only requires seismic data. In particular, we use finite difference to approximate the forward modeling of PDE as a differentiable operator (from velocity map to seismic data) and model its inversion by CNN (from seismic data to velocity map). Hence, we transform the supervised inversion task into an unsupervised seismic data reconstruction task. We also introduce a new large-scale dataset OpenFWI, to establish a more challenging benchmark for the community. Experiment results show that our model (using seismic data alone) yields comparable accuracy to the supervised counterpart (using both seismic data and velocity map). Furthermore, it outperforms the supervised model when involving more seismic data.